Groups, graphs and magic

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1 Groups, graphs and magic Katerina Hristova University of Warwick October 19, 2016 Katerina Hristova (University of Warwick) Groups, graphs and magic October 19, / 16

2 Women in Maths Hypatia Maria Agnesi Sophie Germain Mary Somerville Ada Lovelace Sofia Kovalevskaya Emmy Noether many, many more.. Katerina Hristova (University of Warwick) Groups, graphs and magic October 19, / 16

3 Augusta Ada Byron Born on 10 December 1815 in London, daughter of Lord Byron Got her education in science and mathematics by private tutors amongst which was Mary Somerville Married William King on 8 July 1835 Her husband received the title Earl of Lovelace. Thus, she became Countess of Lovelace Died on 27 November 1852 at the age 36 Katerina Hristova (University of Warwick) Groups, graphs and magic October 19, / 16

4 Ada s Achievements Figure: Reproduction of The Analytical Engine designed by Babbage Worked together with Charles Babbage, who called her enchantress of numbers Developed what is believed to be the first computer algorithm the idea of which was to compute Bernoulli numbers Realised that the engine can not only be used as a computational machine, but also it can be programmed to do other things, i.e. produce music Katerina Hristova (University of Warwick) Groups, graphs and magic October 19, / 16

5 Emmy Noether Born on 23 March 1882 in Erlagen, Germany Got her education in mathematics at the University of Erlagen Taught there for 7 years, moved to University of Gottingen and then to Bryn Mawr College, Pennsylvania Died on 14 April 1935 at the age 53 Katerina Hristova (University of Warwick) Groups, graphs and magic October 19, / 16

6 The significance of Emmy Noether s work Called the most significant creative mathematical genius thus far produced since the higher education of women by Albert Einstein Worked on algebraic invariant theory, Galois theory (in particular on the Inverse Galois problem where she made significant progress) Gave the first definition of a commutative ring in her book Theory of Ideals in Ring Domains (1921) Introduced the concepts of ideals in commutative rings and analysed the ascending chain condition on ideals which led to the discovery of the so called Noetherian rings Worked on representation theory, key discoveries in the representation of an algebra. Also worked noncommutative algebra, topology, physics etc... Katerina Hristova (University of Warwick) Groups, graphs and magic October 19, / 16

7 Group actions on graphs Let G be a group and Γ an oriented graph with vertex set V (Γ) and edge set E(Γ). Defintion G acts on Γ if G acts on V (Γ) and E(Γ). In other words, an action of a group on a graph is a homomorphism φ : G Isom(Γ). Example C 2 acting on a segment by permuting the vertices. We assume throughout that the actions are with no inversion. Katerina Hristova (University of Warwick) Groups, graphs and magic October 19, / 16

8 Two important graphs connected to the group action: The quotient graph is the graph Γ mod G. The fundamental domain of the action is a subgraph X of Γ such that X = Γ mod G. Katerina Hristova (University of Warwick) Groups, graphs and magic October 19, / 16

9 An amalgam of two groups Let G 1, G 2 be groups with presentations G 1 = S 1 R 1 and G 2 = S 2 R 2. Suppose A is another group such that ι 1 : A G 1 and ι 2 : A G 2 are monomorphisms. Defintion The amalgamated product of G 1 and G 2 over A is the group defined by the following presentation: denoted G 1 A G 2. Example S 1, S 2 R 1, R 2, ι 1 (a) = ι 2 (a), for all a A free products are amalgams over the trivial subgroup van Kampen s theorem Katerina Hristova (University of Warwick) Groups, graphs and magic October 19, / 16

10 Main Theorem There exists a one to one correspondence between the following sets: { } } Groups acting on trees with {Groups of the form G fundamental domain a segment 1 A G 2 Katerina Hristova (University of Warwick) Groups, graphs and magic October 19, / 16

11 Examples: The infinite dihedral group D The infinite dihedral group is defined in the following way: D = a, b a 2 = 1, aba = b 1. A simple map sending a x and ab y enables us to rewrite the presentation above as: D = x, y x 2 = y 2 = 1. In particular, we have 2 generators with no relations between them and each of them generates a group of order two. What this means is that: D = Z/2 1 Z/2 = Z/2 Z/2. Katerina Hristova (University of Warwick) Groups, graphs and magic October 19, / 16

12 There exists a tree Γ on which D acts with fundamental domain a segment. Let us find Γ. v 1 e v 2 We know that Stab(v 1 ) = Z/2, Stab(e) = {1} Stab(v 2 ) = Z/2. Thus Γ is the bi-infinite line: v 1 e v 2 Katerina Hristova (University of Warwick) Groups, graphs and magic October 19, / 16

13 Examples: The modular group SL 2 (Z) SL 2 (Z) acts on the upper-half plane H on the left by Mobius transformations: ( a b c d ) z = az + b cz + d, z H. Katerina Hristova (University of Warwick) Groups, graphs and magic October 19, / 16

14 Take the segment γ H with endpoints v 1 = e π/3i and v 2 = i. Using the action above we find: Stab(v 1 ) = Z/6 Stab(e) = Z/2 Stab(v 2 ) = Z/4 Thus, SL 2 (Z) = Z/6 Z/2 Z/4. Katerina Hristova (University of Warwick) Groups, graphs and magic October 19, / 16

15 Generalisations This can be generalised to taking trees instead of segments, and amalgamated products of n groups. The theory behind this beautiful result is called Bass-Serre theory. Katerina Hristova (University of Warwick) Groups, graphs and magic October 19, / 16

16 Thank you for your attention! Katerina Hristova (University of Warwick) Groups, graphs and magic October 19, / 16